I know that the range is the set of output numbers, but I'm not understanding it here.
(1, inf)?

In your above problem, you have given:

f : x |→ (x + 1)² (x ∈ R, x ≤ -2)?

So, we know the domain of $\displaystyle f$ to be $\displaystyle (-\infty, -2]$. We know also know that if $\displaystyle f: A \to B$, then $\displaystyle f^{-1}:B \to A$, and we can therefore conclude that the range of $\displaystyle f^{-1}$ is $\displaystyle (-\infty, -2]$.